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Properties

Label 2-84-7.4-c5-0-3
Degree 22
Conductor 8484
Sign 0.967−0.252i0.967 - 0.252i
Analytic cond. 13.472213.4722
Root an. cond. 3.670453.67045
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 7.79i)3-s + (−32.7 − 56.7i)5-s + (40.6 + 123. i)7-s + (−40.5 − 70.1i)9-s + (122. − 212. i)11-s + 434.·13-s + 590.·15-s + (551. − 954. i)17-s + (1.43e3 + 2.49e3i)19-s + (−1.14e3 − 237. i)21-s + (2.11e3 + 3.66e3i)23-s + (−587. + 1.01e3i)25-s + 729·27-s + 4.96e3·29-s + (4.39e3 − 7.60e3i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.586 − 1.01i)5-s + (0.313 + 0.949i)7-s + (−0.166 − 0.288i)9-s + (0.305 − 0.529i)11-s + 0.713·13-s + 0.677·15-s + (0.462 − 0.801i)17-s + (0.914 + 1.58i)19-s + (−0.565 − 0.117i)21-s + (0.833 + 1.44i)23-s + (−0.187 + 0.325i)25-s + 0.192·27-s + 1.09·29-s + (0.820 − 1.42i)31-s + ⋯

Functional equation

Λ(s)=(84s/2 ΓC(s) L(s)=((0.967−0.252i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(84s/2 ΓC(s+5/2) L(s)=((0.967−0.252i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 8484    =    22⋅3⋅72^{2} \cdot 3 \cdot 7
Sign: 0.967−0.252i0.967 - 0.252i
Analytic conductor: 13.472213.4722
Root analytic conductor: 3.670453.67045
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ84(25,⋅)\chi_{84} (25, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 84, ( :5/2), 0.967−0.252i)(2,\ 84,\ (\ :5/2),\ 0.967 - 0.252i)

Particular Values

L(3)L(3) ≈\approx 1.56939+0.201627i1.56939 + 0.201627i
L(12)L(\frac12) ≈\approx 1.56939+0.201627i1.56939 + 0.201627i
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(4.5−7.79i)T 1 + (4.5 - 7.79i)T
7 1+(−40.6−123.i)T 1 + (-40.6 - 123. i)T
good5 1+(32.7+56.7i)T+(−1.56e3+2.70e3i)T2 1 + (32.7 + 56.7i)T + (-1.56e3 + 2.70e3i)T^{2}
11 1+(−122.+212.i)T+(−8.05e4−1.39e5i)T2 1 + (-122. + 212. i)T + (-8.05e4 - 1.39e5i)T^{2}
13 1−434.T+3.71e5T2 1 - 434.T + 3.71e5T^{2}
17 1+(−551.+954.i)T+(−7.09e5−1.22e6i)T2 1 + (-551. + 954. i)T + (-7.09e5 - 1.22e6i)T^{2}
19 1+(−1.43e3−2.49e3i)T+(−1.23e6+2.14e6i)T2 1 + (-1.43e3 - 2.49e3i)T + (-1.23e6 + 2.14e6i)T^{2}
23 1+(−2.11e3−3.66e3i)T+(−3.21e6+5.57e6i)T2 1 + (-2.11e3 - 3.66e3i)T + (-3.21e6 + 5.57e6i)T^{2}
29 1−4.96e3T+2.05e7T2 1 - 4.96e3T + 2.05e7T^{2}
31 1+(−4.39e3+7.60e3i)T+(−1.43e7−2.47e7i)T2 1 + (-4.39e3 + 7.60e3i)T + (-1.43e7 - 2.47e7i)T^{2}
37 1+(−1.22e3−2.11e3i)T+(−3.46e7+6.00e7i)T2 1 + (-1.22e3 - 2.11e3i)T + (-3.46e7 + 6.00e7i)T^{2}
41 1+3.66e3T+1.15e8T2 1 + 3.66e3T + 1.15e8T^{2}
43 1+7.19e3T+1.47e8T2 1 + 7.19e3T + 1.47e8T^{2}
47 1+(1.63e3+2.83e3i)T+(−1.14e8+1.98e8i)T2 1 + (1.63e3 + 2.83e3i)T + (-1.14e8 + 1.98e8i)T^{2}
53 1+(−1.51e3+2.61e3i)T+(−2.09e8−3.62e8i)T2 1 + (-1.51e3 + 2.61e3i)T + (-2.09e8 - 3.62e8i)T^{2}
59 1+(−2.57e4+4.45e4i)T+(−3.57e8−6.19e8i)T2 1 + (-2.57e4 + 4.45e4i)T + (-3.57e8 - 6.19e8i)T^{2}
61 1+(−6.65e3−1.15e4i)T+(−4.22e8+7.31e8i)T2 1 + (-6.65e3 - 1.15e4i)T + (-4.22e8 + 7.31e8i)T^{2}
67 1+(1.54e4−2.67e4i)T+(−6.75e8−1.16e9i)T2 1 + (1.54e4 - 2.67e4i)T + (-6.75e8 - 1.16e9i)T^{2}
71 1+4.18e4T+1.80e9T2 1 + 4.18e4T + 1.80e9T^{2}
73 1+(1.73e4−2.99e4i)T+(−1.03e9−1.79e9i)T2 1 + (1.73e4 - 2.99e4i)T + (-1.03e9 - 1.79e9i)T^{2}
79 1+(3.87e4+6.71e4i)T+(−1.53e9+2.66e9i)T2 1 + (3.87e4 + 6.71e4i)T + (-1.53e9 + 2.66e9i)T^{2}
83 1−1.00e5T+3.93e9T2 1 - 1.00e5T + 3.93e9T^{2}
89 1+(2.03e4+3.53e4i)T+(−2.79e9+4.83e9i)T2 1 + (2.03e4 + 3.53e4i)T + (-2.79e9 + 4.83e9i)T^{2}
97 1−1.40e5T+8.58e9T2 1 - 1.40e5T + 8.58e9T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−13.26027166012384193637313671824, −11.79122298824871458356527287760, −11.69238872535378780076750628609, −9.886892997296700479002632918971, −8.830356180626795159502430988931, −7.930966958132089606405990448490, −5.91514265393052544770492906593, −4.94664991579940228256524848705, −3.46041546340850499760503935043, −1.05633571407992533871828736756, 0.990555564616122450622168838021, 3.09244846324587534226707969096, 4.64343726847038328327050505178, 6.62331135394178576187184158284, 7.21260997944316618923009705267, 8.509879180809295599831043552427, 10.34928873525461886716739690646, 11.02842325538722746114786060879, 12.05291833619398836171219747383, 13.33064502462539439191746420152

Graph of the ZZ-function along the critical line